Stability and asymptotic behaviour of one-dimensional solutions in cylinders

2025/09/11 by De Marchis, Francesca, Mazzuoli, Lisa, Pacella, Filomena · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.09648

Abstract

We consider positive one-dimensional solutions of a Lane-Emden relative Dirichlet problem in a cylinder and study their stability/instability properties as the energy varies with respect to domain perturbations. This depends on the exponent p >1 of the nonlinearity and we obtain results for p close to 1 and for p large. This is achieved by a careful asymptotic analysis of the one-dimensional solution as p → 1 or p → ∞, which is of independent interest. It allows to detect the limit profile and other qualitative properties of these solutions.

Citations

• Nontrivial solutions to the relative overdetermined torsion problem in a cylinder
• Semilinear equations in bounded cylinders: Morse index and bifurcation from one-dimensional solutions
• Energy stability for a class of semilinear elliptic problems
• Asymptotic profile of positive solutions of Lane-Emden problems in\n dimension two

Cited by

• Bifurcating domains for an overdetermined eigenvalue problem in cylinders

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